The Tests Behind the Story
The full statistical work behind the blog post. Formal hypothesis tests on the 4–5% claim, stationarity and mean-reversion analysis on REER, and an interactive scenario tool that lets you dial inflation assumptions and watch the implied USD/INR fair value update in real time. Three sections — Nominal, Real, Scenario.
← Back to the articleNominal: Testing the 4–5% Rule
Hypothesis tests run on the floating-regime sample (n = 32 calendar years, December-to-December, 1994 through 2025). Significance at α = 0.05.
Mean YoY
CAGR
95% CI for μ
Years in [4%, 5%]
Years appreciated
40 Years of USD/INR — Level and Log-Linear Fit
Monthly USD/INR close, Jan 1986 → May 2026. Dashed line is the OLS log-linear fit over the floating-regime period (slope = 3.34%/yr). Shaded band marks the pre-1993 pegged regime.
Calendar-Year Depreciation, with the 4–5% Hypothesis Band
Each bar is one calendar year's % change in USD/INR. Green band = the 4–5% hypothesis zone. Red dashed line = sample mean (3.58%). Only 2005 (4.03%) lands inside the band.
Formal Hypothesis Tests
| Test | Null hypothesis | Statistic | p-value | Decision @ α=5% |
|---|---|---|---|---|
| One-sample t-test | μ = 4.0% | t = −0.34 | 0.738 | Cannot reject |
| One-sample t-test | μ = 4.5% | t = −0.73 | 0.468 | Cannot reject |
| One-sample t-test | μ = 5.0% | t = −1.13 | 0.267 | Cannot reject |
| Wilcoxon signed-rank | median = 4% | W = 213 | 0.350 | Cannot reject |
| Wilcoxon signed-rank | median = 5% | W = 188 | 0.160 | Cannot reject |
| TOST equivalence | μ ∈ [4%, 5%] | — | pTOST = 0.631 | Cannot confirm equivalence |
| Shapiro–Wilk | YoY ∼ Normal | W = 0.96 | 0.266 | Normality holds |
| Jarque–Bera | YoY ∼ Normal | JB = 2.99 | 0.225 | Normality holds |
| Mann–Kendall trend | No monotonic trend | Z = −0.02 | 0.987 | No trend |
| OLS slope (YoY ∼ year) | slope = 0 | β = 0.004%/yr | 0.979 | Flat |
| ADF (log USD/INR) | Unit root present | τ = −0.31 | 0.924 | I(1) — non-stationary in levels |
| ADF (Δ log USD/INR) | Unit root present | τ = −5.43 | <0.0001 | Stationary in log-returns |
| ADF (YoY series) | Unit root present | τ = −3.29 | 0.015 | Stationary |
The t-tests fail to reject 4%, 4.5%, or 5% because the 95% CI for the true mean — [1.01%, 6.14%] — is wide enough to contain all three. But the TOST equivalence test, which would positively confirm that μ lies in [4%, 5%], also fails (pTOST = 0.63). The level series is I(1) but log-returns are stationary — consistent with random-walk-with-drift behaviour. The drift parameter, estimated at 3.34% per year, is the most honest single-number answer to "what does the rupee do per year on average."
Structural Break Tests (Welch t)
| Candidate break year | Pre-break mean | Post-break mean | Welch t | p-value | Verdict |
|---|---|---|---|---|---|
| 2000 | 5.69% | 3.09% | 1.05 | 0.315 | No break |
| 2003 | 4.91% | 3.05% | 0.83 | 0.412 | No break |
| 2008 (GFC) | 1.83% | 4.94% | −1.27 | 0.215 | No break |
| 2013 (taper) | 3.32% | 3.95% | −0.27 | 0.790 | No break |
| 2018 | 3.29% | 4.43% | −0.54 | 0.593 | No break |
Sub-period CAGRs
| Period | Years | CAGR | Context |
|---|---|---|---|
| 1993–2000 | 7 | 5.84% | Post-unification, Asian crisis |
| 2000–2010 | 10 | −0.43% | "Lost decade" for the depreciation thesis |
| 2010–2020 | 10 | 5.03% | Taper tantrum + USD strength |
| 2020–2025 | 5 | 4.24% | Post-pandemic, recent acceleration |
| 1993–2025 | 32 | 3.34% | Full floating era |
Regime Comparison
| Regime | n | Mean | Median | Std | Range |
|---|---|---|---|---|---|
| Pegged (1987–1993) | 7 | 14.04% | 12.34% | 14.81% | −2.8% to +42.2% |
| Floating (1994–2025) | 32 | 3.58% | 2.61% | 7.12% | −10.7% to +23.4% |
Real: Trend, Stationarity, and Mean Reversion
Tests run on monthly BIS Broad REER for India (FRED: RBINBIS), n = 387 observations, January 1994 → March 2026.
Long-run mean
Trend slope
AR(1) φ (detrended)
Mar 2026 reading
REER Level, Trend, and ±σ Bands
Solid blue: REER monthly. Dashed: OLS linear trend. Shaded corridors: ±1σ (68% historical coverage) and ±2σ (96%). Red dot: implied May 2026 reading.
Deviation from Trend
Blue bars = REER above trend, red bars = below trend. Dotted lines mark ±1σ and ±2σ thresholds.
Stationarity & Trend Tests
| Test | Null hypothesis | Statistic | p-value | Verdict |
|---|---|---|---|---|
| ADF on level | Unit root | τ = −2.64 | 0.086 | Borderline non-stationary |
| ADF on detrended residuals | Unit root | τ = −4.61 | 0.0001 | Strongly stationary around trend |
| KPSS (level, regression=c) | Stationary | 1.98 | 0.010 | Reject — not mean-stationary |
| KPSS (trend, regression=ct) | Trend-stationary | 0.07 | 0.100 | Cannot reject — trend-stationary |
| Mann–Kendall | No trend | Z = 14.67 | <0.0001 | Significant upward trend |
| OLS slope | slope = 0 | β = 0.408/yr | <0.0001 | Trend confirmed |
ADF + KPSS together give the clearest read: REER is trend-stationary, not mean-stationary. The level wanders around an upward-drifting trend, and deviations from that trend are pulled back. The right mental model is "wanders around an upward-sloping fair value line," not "wanders aimlessly forever."
Speed of Mean Reversion — AR(1)
| Specification | φ | p-value | Half-life |
|---|---|---|---|
| Deviation from trend | 0.935 | <10⁻¹⁰⁰ | 10.3 months |
| Deviation from long-run mean | 0.965 | <10⁻¹⁰⁰ | 19.4 months |
Persistence — Hurst Exponent
| Series | Hurst H | Interpretation |
|---|---|---|
| REER level | 0.91 | Highly persistent (reflects underlying trend) |
| Detrended REER | 0.86 | Persistent — short-term momentum, slow reversion |
Variance Ratio Tests (Lo-MacKinlay)
| q (months) | VR | z | p-value | Interpretation |
|---|---|---|---|---|
| 2 | 1.125 | 2.46 | 0.014 | Mild momentum at 1mo |
| 4 | 1.052 | 0.55 | 0.582 | Neutral |
| 12 | 1.053 | 0.28 | 0.781 | Neutral |
| 24 | 0.704 | −1.06 | 0.289 | Mean-reverting at 2-yr |
| 60 | 0.281 | −1.60 | 0.110 | Strong reversion at 5-yr |
Current Standing
| Reading | REER | vs Long-run mean | vs Trend line |
|---|---|---|---|
| March 2026 (last hard data) | 91.05 | −0.62σ | −2.60σ |
| Implied May 2026 (after INR 93.5 → 96.1) | ~88.75 | −1.05σ | −3.22σ |
Scenario Tool: Where Could REER Land — and What Does It Imply for USD/INR?
Take the last hard reading (March 2026, REER 91.05). Apply your assumed 2-month inflation differential and the known +2.83% nominal USD/INR move to get the implied May 2026 REER. Then project forward 12 months using the empirical 10.3-month half-life of mean reversion. Output: implied USD/INR fair value at May 2027.